To summarize, we have for the first time systematically realized the HOM interference between the hyper-entangled two photons. It is greatly interesting to extend our work to high-dimensional region by exploring the HOM effect with hyper-entanglement in (two-dimensional) polarization and other (high-dimensional) DoF such as OAM. Benefitted from the recently developed technologies of high-dimensional entanglement preparation [1]}, [2]} and measurement methods [3]}, [4]}, the HOM interference of such hyper-entangled two photons is expected in future. Our approach enables to directly measure the exchange symmetric and anti-symmetric phases, demonstrating an interesting and significant application of the hyper-entangled two-photon interference. More applications of the HOM interference in the more DoFs would be expected in quantum network [5]}, [6]}, quantum computation [7]} and high-dimensional quantum entanglement and processing [8]}, [9]}, [10]}. Our results show that expanding the Hilbert space by introducing the extra DoFs will open new opportunities for quantum information, such as turning the external and unmeasurable exchange phase in one DoF to an internal and measurable phase in another DoF. More DoFs not only contribute to high-dimensional quantum information processing [10]}, but also excite novel functions in quantum applications such as alignment-free quantum communications [12]} and complete Bell state measurement [13]}.
